TIME-FREQUENCY ESTIMATES FOR PSEUDODIFFERENTIAL

TIME-FREQUENCY ESTIMATES FOR PSEUDODIFFERENTIAL OPERATORS ´ ´ BENYI ´ ARP AD AND KASSO A. OKOUDJOU∗ Abstract. We present several boundedness results for linear and multilinear pseudodifferential operators on modulation spaces.

1. Introduction The recent years have seen an increasing interest in the theory of modulation spaces. The family of modulation spaces appear naturally in the study of certain functions and operators, especially when one is interested in both the time and frequency description of such objects. They constitute a family of Banach spaces of distributions that behave very much like the Besov spaces: the dilation in the definition of Besov spaces is essentially replaced by frequency shifts. In this paper, we present certain results and techniques regarding the boundednes of linear and multilinear operators acting on these spaces. Our goal is to provide a story of estimates, known and unknown, on modulation spaces. We would like to convey to the reader that such estimates are natural substitutes when other classical function space estimates fail. Moreover, general estimates on modulation spaces translate-via embeddings- into estimates on Lebesgue, Besov, or Sobolev spaces. In telling our story of estimates, we choose a non-uniform approach by jumping from linear to multilinear estimates. Several topics presented here might seem unrelated when read separately. The unifying theme, however, is the general framework of modulation spaces that led us to ask certain questions in the first place once previous questions have been answered in a satisfactory manner. Our paper is organized as follows. In Section 2 we set the notations and definitions that will be used throughout this paper. We also define the modulation spaces and collect some of their properties that will be needed later on. In Section 3 we present an overview of estimates for various operators. To the best of our knowledge, some of these estimates or techniques do not appear elsewhere in the literature.

Date: April 6, 2007. 2000 Mathematics Subject Classification. Primary 47G30; Secondary 42B35, 35S99. Key words and phrases. Linear operators, multilinear operators, pseudodifferential operators, multipliers, modulation spaces, moderate weights, short-time Fourier transform, time-frequency analysis. ∗ Research partially supported by an Erwin Schrödinger Junior Fellowship, and by NSF grant DMS-0139261. 1

´ BENYI ´ A. AND K. A. OKOUDJOU

2

2. Preliminaries 2.1. General notation. Translation and modulation of a function f with domain Rd are defined, respectively, by Tx f (t) = f (t − x) and My f (t) = e2πiy·t f (t). R The Fourier transform of f ∈ L1 is fˆ(ω) = Rd f (t) e−2πit·ω dt, ω ∈ Rd . The Fourier transform is an isomorphism of the Schwartz space S onto itself, and extends to 0 the space S of tempered distributions by duality. The inverse Fourier transform is fˇ(x) = fˆ(−x). R The inner product of two functions f, g ∈ L2 is hf, gi = Rd f (t)g(t) dt, and its 0 extension to S × S will be also denoted by h·, ·i. The Short-Time Fourier Transform (STFT) of a function f with respect to a window g is Z e−2πiy·t g(t − x) f (t) dt,

Vg f (x, y) = hf, My Tx gi = Rd

0

whenever the integral makes sense. If g ∈ S and f ∈ S then Vg f is a uniformly continuous function on R2d . One important technical tool is the extended isometry property of the STFT [11, (14.31)]: If φ ∈ S, kφkL2 = 1, then (1)

hf, hi = hVφ f, Vφ hi,

0

∀ f ∈ S , h ∈ S.

Given a strictly positive function ν on R2d , we let Lp,q ν be the spaces of measurable functions f (x, y) for which the weighted mixed norms q/p 1/q Z Z p p p,q kf kLν = |f (x, y)| ν(x, y) dx dy Rd

Rd

2d p 2d are finite. If p = q, we have Lp,p ν (R ) = Lν (R ), a weighted Lebesgue space.

2.2. Weight functions. Given s ≥ 0, a positive, continuous, and symmetric function ν is called an s-moderate weight if there exists a constant C > 0 such that (2)

∀ x, y ∈ Rd ,

ν(x + y) ≤ C (1 + |x|2 )s/2 ν(y).

For example, ν(x) = (1 + |x|2 )t/2 is s-moderate exactly for |t| ≤ s. If ν is s-moderate, then by manipulating (2) we see that 1 1 ≤ C (1 + |x|2 )s/2 , ν(x + y) ν(y) hence, 1/ν is also s-moderate (with the same constant). In the sequel we let ωs (x) = (1 + |x|2 )s/2 for s ≥ 0. 2.3. Modulation spaces. Definition 1. Given 1 ≤ p, q ≤ ∞, and given a window function g ∈ S, and an sd moderate weight ν defined on R2d , the modulation space Mp,q = Mp,q ν ν (R ) is the 0 space of all distributions f ∈ S for which the following norm is finite: Z Z q/p 1/q p p (3) kf kMp,q = |Vg f (x, y)| ν(x, y) dx dy = kVg f kLp,q , ν ν Rd

Rd

TIME-FREQUENCY ESTIMATES

3

with the usual modifications if p and/or q are infinite. If ν = 1, we will simply write p,q Mp,q for the modulation space Mp,q ν . If ν(x, y) = ωs (x), we will simply write Ms . p p,p Moreover, when p = q, we will write Mν for the modulation space Mν . The definition is independent of the choice of the window g in the sense of equivalent norms. We refer to [7, 11] and the references therein for more details about modulation spaces. The definition above quantifies both the time and frequency contents of a function or distribution. Although not completely correct, one can think of f ∈ Mp,q as being represented by the statement “f ∈ Lp and fˆ ∈ Lq ”; for a rigorous comparison of modulation spaces and Fourier-Lebesgue spaces see [10]. Fore more embeddings between modulation spaces and other function spaces, see [10, 12, 14, 17, 21]. Remark 1. For p = q = 2, if ν ≡ 1 then M2ν = L2 ; if ν(x, y) = (1 + |x|2 )s/2 then M2ν = L2ωs = L2(s) , a weighted L2 -space; if ν(x, y) = (1 + |y|2 )s/2 then M2ν = H s , the standard Sobolev space, and if ν(x, y) = (1 + |x|2 + |y|2 )s/2 then M2ν = L2(s) ∩ H s . 3. Estimates A k-linear pseudodifferential operator is defined à priori through its (distributional) symbol σ to be the mapping Tσ from the k-fold product of Schwartz spaces S×· · ·×S 0 into the space S of tempered distributions given by the formula

(4)

Tσ (f1 , . . . , fk )(x) Z = σ(x, ξ1 , . . . , ξk ) fˆ1 (ξ1 ) · · · fˆk (ξk ) e2πix·(ξ1 +···+ξk ) dξ1 · · · dξk , Rkd

for f1 , . . . , fk ∈ S. 1-linear operators are simply called linear, 2-linear operators are called bilinear, and so on. The pointwise product f1 · · · fk corresponds to the case σ ≡ 1. We start our story of estimates with a classical class of symbols. 3.1. The Calder´ on-Vaillancourt class. Assume that σ ∈ L∞ and consider the linear operator L1 (f ) = σf . Clearly, this operator is bounded on all Lp spaces, 1 ≤ p ≤ ∞. Hölder’s inequality tells us that the bilinear operator L2 (f, g) = σf g is bounded from Lp × Lq into Lr if the exponents p, q are larger than 1 and satisfy the relation 1/p + 1/q = 1/r. The norms of operators L1 , L2 coincide with kσkL∞ . These cases correspond to pseudodifferential operators with symbol σ(x, ξ) = σ(x) independent of the frequency variable ξ. A much more interesting class of operators is defined on the frequency side by [ T (f ) = σ fˆ. In this case, T coincides with a Fourier multiplier with symbol σ(x, ξ) = σ(ξ) independent of the space variable x. Using Plancherel’s identity, we see that T is bounded from L2 into L2 if and only if σ ∈ L∞ . It is not clear, however, how one would approach the boundedness of T on Lp for p 6= 2. If one could obtain another Lp0 estimate, then interpolation would give boundedness for all exponents in the range (2, p0 ) or (p0 , 2). A naive approach would be, for example, the following: kT (f )kL∞ = k(mfˆ)ˇkL∞ ≤ kmfˆkL1 ≤ kmkL∞ kfˆkL1 .


4

But this is perhaps the most that we can accomplish this way, since, in general, we do not have any good control on the L1 norm of the Fourier transform of a function. The same approach would have failed if we would have started with a different exponent p0 . Nevertheless, we proved that the operator T is bounded from L2 → L2 and c1 → L∞ . The natural question then is whether there are any “intermediate” spaces L on which the operator remains bounded. The questions above become yet more difficult if we modify the operators and allow the symbols to be both x and ξ dependent. Consider then the classical symbol class 0 S0,0 consisting of those σ which satisfy estimates of the form (5)

|∂xα ∂ξβ σ(x, ξ)| ≤ Cα,β ,

∀ α, β ≥ 0.

A classical result of Calderón and Vaillancourt [6] asserts that the corresponding linear pseudodifferential operator Tσ is bounded on L2 . Notice that these operators are generally unbounded on Lp for p 6= 2 [1]. In the bilinear case, however, the analogous class of symbols which satisfy the conditions (6)

|∂xα ∂ξβ ∂ηγ σ(x, ξ, η)| ≤ Cα,β,γ ,

∀ α, β, γ ≥ 0,

does not necessarily yield bounded operators from L2 × L2 into L1 , unless additional size conditions are imposed on the symbols; see [5]. Nevertheless, a Calderón– Vaillancourt-like condition (6) does yield boundedness from L2 × L2 into the modulation space M1,∞ that contains L1 . Indeed, considering this problem in the framework of modulation spaces we find the desired answer for general k-linear pseudodifferential operators; see [3] for the proof and further details. Theorem 1. If σ ∈ M∞,1 (R(k+1)d ), then the k-linear pseudodifferential operator Tσ defined by (4) extends to a bounded operator from Mp1 ,q1 × · · · × Mpk ,qk into Mp0 ,q0 when p11 + · · · + p1k = p10 , q11 + · · · + q1k = k − 1 + q10 , and 1 ≤ pi , qi ≤ ∞ for 0 ≤ i ≤ k. All the questions we asked above can be answered from this theorem, by using the 0 embeddings S0,0 ⊂ M∞,1 , L1 ⊂ M1,∞ , M2,2 = L2 , and choosing k = 1 respectively k = 2. It is worth pointing out that there is an extensive literature on the continuity properties of (linear) pseudodifferential operators on modulation spaces; see [9, 13, 18, 21]. 3.2. The Hilbert transform and related multipliers. Closely related to the 0 linear operators with symbols in the class S0,0 is the Hilbert transform defined by Z 1 f (x − t) Hf (x) = lim dt. π →0 |t|> t H is a Fourier multiplier operator (7)

d = mfb, Hf

where m(ξ) = −isgn ξ. The multiplier m is bounded and its derivatives, which exist everywhere except at the origin, are also bounded. Thus, m “almost” belongs to


5

0 . Yet, its behavior does not follow the pattern we saw in the previous the class S0,0 subsection: the Hilbert transform is bounded on all Lp spaces, 1 < p < ∞. This is due to the fact that H enjoys additional cancellation properties, and as such it can be treated in the context of Calderón-Zygmund theory; m is known to be a particular example of a Hörmander-Mihlin multiplier as well. Nevertheless, m does follow the pattern of the Calderón-Vaillancourt class on modulation spaces. In fact, a larger class of multipliers extending the Hilbert transform enjoys this property of boundedness on modulation spaces. Let b > 0 and c = (cn )n∈Z be a bounded sequence of complex numbers. The operators Hb,c are defined by

(8)

b [ H b,c f = mb,c f ,

with Fourier multipliers (9)

mb,c = −2i

+∞ X

cn χ(bn,b(n+1)) ;

n=−∞

χ(a,b) denotes the characteristic function of the real interval (a, b). It is easy to see that H = 12 Hb,c for any b > 0 and c = (cn )n∈Z , with cn = 1 for n ≥ 0 and cn = −1 for n < 0. The following result was proved in [2]. Theorem 2. The operators Hb,c are bounded from Mp,q (R) into Mp,q (R) for 1 < p < ∞, 1 ≤ q ≤ ∞ with a norm estimate kHb,c f kMp,q ≤ C kck∞ kf kMp,q for some constant depending only on b, p, and q. In particular, the Hilbert transform H is bounded on Mp,q for 1 < p < ∞ and 1 ≤ q ≤ ∞ Remark 2. Interestingly enough, the multiplier operators Hb,c are not bounded in general on Lp spaces, except when p = 2. As such, they resemble the behavior 0 investigated in the previous subsection: the modulation spaces are of the class S0,0 the “appropriate” spaces on which to study their boundedness. See [2] for some extensions of Theorem 2. In [8], a slightly more general class of multipliers as the one exposed in Theorem 2 was shown to be bounded on modulation spaces. 3.3. Fourier multipliers and evolution PDEs. A strong motivation for the development of a theory of pseudodifferential operators is provided by the fact that pseudodifferential operators generalize classical partial differential operators with variable coefficients. As such, it is not surprising that they appear in the study of solutions of certain partial differential equations. An extensive amount of research in the area of partial differential equations has been devoted in recent years to study the well-posedness or solvability of various dispersive equations; see [19], also [20]. In the remaining of this subsection we would like to explore two such classical evolution equations: the Schrödinger and wave ones. We will see how to approach their behavior in the context of modulation spaces, as well as the connections with the operators discussed above.


6

The Schrödinger equation iut − ∆u = 0, for example, with u a complex valued function in Rd × R, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. This equation can be perturbed in many ways, mainly by adding a potential or an obstacle, and the resulting equations arise as models from several areas of physics. Clearly, the solution of the corresponding Cauchy problem with initial data f is given by a Fourier multiplier operator: Z (10) u(x, t) = Tt f (x) = mt (ξ)e2π ix·ξ fˆ(ξ)dξ, Rd it|ξ|2

where mt (ξ) = e . It is known that Tt is bounded on Lp (Rd ) only for p = 2 [15]. Therefore, it is natural to ask whether this Fourier multiplier operator is of Calderón-Vaillancourt type on modulation spaces. Similarly, the wave Cauchy problem ∂tt u−∆u = 0, u(x, 0) = f (x), ∂t u(x, 0) = g(x) has a solution represented by a sum of two Fourier multiplier operators. In this case u(x, t) = Tt1 f (x) + Tt2 g(x), where, like in (10), the multipliers are given by m1t (ξ) = cos t|ξ|, and m2t (ξ) = sin(t|ξ|)/|ξ|. Again, it is known in this case that Tti , i = 1, 2, are bounded on Lp (Rd ) only when p = 2 and d ≥ 1 or when 1 < p < ∞ and d = 1 [15, 16]. In a recent work [4], combining tools from Littlewood-Paley theory and timefrequency analysis, it has been proved that a large family of Fourier multipliers whose α symbols are given by unimodular functions ei|ξ| for a specific range of α, are bounded on all modulation spaces. It is worth pointing out that these operators are in general unbounded on other classical function spaces such as the Lebesgue spaces. 3.4. Strichartz-type estimates. The solution of the linear Schrödinger equation iut − ∆u = 0 with initial condition u(x, 0) = f (x) is given by a t-parameter Fourier multiplier (10) u(x, t) = Tt f (x), typically written as u(x, t) = eit∆ f (x). The classical Strichartz estimates for the Schrödinger equation give Lqt Lrx bounds on the solution u(x, t) in terms of the initial condition f . Here, ku(x, t)kLqt Lrx = kku(·, t)kLrx kLqt . The pair of exponents (q, r) is called Schr¨ odinger admissible if q, r ≥ 2, (q, r, d) 6= (2, ∞, 2) and 1/q + d/2r = d/4. For such a pair we have (11)

keit∆ f kLqt Lrx . kf kL2x .

In particular, for the admissible pair (q ∗ , q ∗ ), q ∗ = 2(d + 2)/d, (11) gives (12)

keit∆ f kLq∗ (Rd ×R) . kf kL2 (Rd ) . ∗

∗

∗

But, since q ∗ > 2, we have Lq ⊂ Mq ,q , and L2 = M2,2 , therefore from (12) we obtain a Strichartz estimate for a pair of modulation spaces (13)

keit∆ f kMq∗ ,q∗ . kf kM2,2 .

It is natural to ask then if there are other Schr¨ odinger admissible quadruples (p, q, r, s) such that (14)

keit∆ f kMp,q (x,t) . kf kMr,s (x) .


7

Similarly, it would be interesting to know for which Schr¨ odinger admissible triples (p, q, r) we have an estimate like (15)

keit∆ f kMp,q (x,t) . kf kLr (x) . 0

Note that from the embeddings Lp ⊆ Mp,p , 1 ≤ p ≤ 2 and Lp ⊆ Mp,p , 2 ≤ p ≤ ∞, we immediately get the following spatial estimates for triples (p, p0 , p0 ) respectively (p, p, p0 ): 1 1 keit∆ f kMp,p0 (x) . |t|−d( 2 − p ) kf kLp0 (x) , 1 ≤ p ≤ 2 and 1 1 keit∆ f kMp,p (x) . |t|−d( 2 − p ) kf kLp0 (x) , 2 ≤ p ≤ ∞. However, only from these inequalities we cannot conclude estimates of the form (15). Indeed, unlike the Lebesgue spaces, the modulation spaces are not lattice spaces, that is, it is not true that if |f | ≤ |g| and g ∈ Mp,q then f ∈ Mp,q . Following ideas from the classical Schrödinger estimates on mixed Lebesgue spaces, we would like to make use of the scaling properties of the equation to arrive to a possible definition of admissibility for quadruples in (14). A similar condition will then give admissibility for triples in (15). Unfortunately, the scaling of the STFT with respect to a window function g gets corrupted, therefore we need to correct it by introducing a new parameter in the definition of the STFT. For µ > 0 and a given window g, the d-dimensional µ-STFT is defined by Z z−x −d e−2πiy·z g( (16) Vg;µ f (x, y) = µ ) f (z) dt. µ Rd Let fλ (z) = f (λz). It is easy to see that y Vg;µ fλ (x, y) = Vg;µλ f (λx, ) λ and

1

1

kVg;µ fλ kLp,q = λd( q − p ) kVg;µλ f kLp,q . p,q through ] It is natural then to define the spaces M (17)

p,q . kf (x)kM p,q = sup kVg;µ f kL ^

µ>0

p,q ⊂ Mp,q . For the smaller spaces M p,q ] ] Clearly, by letting µ = 1 in (17), we have M we have a “good” scaling property 1

1

d( q − p ) kfλ kM kf kM p,q = λ p,q . ^ ^

This takes care of the scaling of the initial condition. We now take a similar route to deal with the scaling of the solution u(x, t). In particular, we would like to investigate the mixed Lebesgue norms of appropriate (µ1 , µ2 )-STFT of scaled solutions uλ (x, t) = u(x/λ, t/λ2 ). Here, we distinguish between the positive parameters µ1 and µ2 that refer to the distinct homogeneities in x respectively t. With g = g1 ⊗ g2 , a similar computation as above gives 1

1

kVg;µ1 ,µ2 uλ kLp,q = λ(d+2)( p − q ) kVg;µ1 /λ,µ2 /λ2 ukLp,q .


8

In particular, by denoting (18)

ku(x, t)kMp,q = sup kVg;µ1 ,µ2 ukLp,q , µ1 ,µ2 >0

we obtain the scaling property 1

1

kuλ kMp,q = λ(d+2)( p − q ) kukMp,q . We know that if u is a solution of the linear Schrödinger equation with initial condition f , then uλ is also a solution for the problem with initial condition fλ . Therefore, in order for an estimate (19)

keit∆ f kMp,q (x,t) . kf kM r,s (x) ]

to hold, the quadruple (p, q, r, s) must satisfy the admissibility condition 1 1 1 1 d( − ) = (d + 2)( − ). s r p q Admissibility for triples (p, q, r) in the analogue of (15) with Mp,q -norms on the left hand-side is d 1 1 = (d + 2)( − ). r q p Note, however, that the admissibility condition for quadruples is not sufficient. For example, the quadruple (2, 2, 2, 2) is admissible, yet (19) does not hold. This is a simple consequence of Plancherel’s theorem: ku(t, ·)kL2x = kf kL2x . Unfortunately, as it was pointed out to us by K. Gröchenig, there is a flaw in this p,q (or Mp,q ) through ] approach. While introducing the modified modulation spaces M the parametrized STFT makes sense, our definition might give only the trivial {0} space in certain situations. For example, for p = q = 2, kVg,µ fλ kL2 = kVg,µλ f kL2 , and kVg,µ f kL2 = kf kL2 kgµ kL2 = kf kL2 kgkL2 µ−d/2 . This explains why certain quadruples, although admissible, do not yield the expected boundedness. It is not clear to these authors what modifications, if any, are needed to our arguments, or what additional restrictions to require on the quadruples that would guarantee estimates of the form (14). It may also well be the case that the quadruple (q ∗ , q ∗ , 2, 2) (which is admissible) is in fact the only one for which a modulationStrichartz type estimate holds. 3.5. Derivative estimates. We end our overview of estimates on modulation spaces with the following boundedness results about partial differential operators. The results we present in this subsection could be obtained also from the “lifting” property of modulation spaces; see [7, Remark 6.3] and [22, Corollary 3.3]. Moreover, the “lifting” property of the modulation spaces was used in [7, 22] to give certain modulation space estimates for fractional derivatives. Let ν(x, y) = νa,b (x, y) = (1 + |x|2 )a/2 (1 + |y|2 )b/2 . p,q We denote the weighted modulation space Mp,q ν by Ma,b . The weighted Lebesgue space Lpνa,0 is denoted by Lp(a) .


9

Theorem 3. If 1 ≤ r ≤ p, 1 ≤ s ≤ q, a > d(1/r − 1/p), and b > d( 1s − 1q ), then k

∂αf kMr,s ≤ C(p, q, r, s, α, d)kf kMp,q a,b ∂xα

for all multiindices α such that |α| < b − d( 1s − 1q ). Proof. Note that for a given window g, we have ∂αf ∂α |α| )(u, v) = (−1) hf, Mv Tu gi. ∂xα ∂xα Using Leibniz’s rule to compute the derivative of the product My Tx g, and using the ∂ α−γ g notation gα−γ = ∂x α−γ , we obtain Vg (

Vg (

X ∂αf |α| )(u, v) = (−1) cα,γ (2πi)|γ| v γ Vgα−γ f (u, v). α ∂x |γ|≤|α|

Therefore, kVg (

X ∂αf r )(·, v)k ≤ C(α) |v||γ| kVgα−γ f (·, y)kLr . L ∂xα |γ|≤|α|

Using the conditions on the parameters a, r, p and Hölder’s inequality, we can write Z r kVgα−γ f (·, v)kLr = |Vgα−γ f (u, v)|r (1 + |u|2 )ra/2 (1 + |u|2 )−ra/2 du (20)

≤ C(r, p, d)kVgα−γ f (·, v)krLp . (a)

(21) Integrate now with respect to y to obtain X ∂αf k(1 + |v|2 )|γ|/2 kVgα−γ f (·, v)kLp(a) kLs . kVg ( α )kLr,s ≤ C(p, r, α, d) ∂x |γ|≤|α|

Using now the conditions on b, s, q, a similar argument as above using Hölder’s inequality allows us to conclude the inequality we wanted to prove. X ∂αf Corollary 1. Let P (D)(f ) = aα α be a partial differential operator with con∂x |α|≤k

r,s for a, b, p, q, r, s as in Theorem 3, and stant coefficients. Then P (D) : Mp,q a,b → M 1 1 |k| < b − d( s − q ).

4. Acknowledgment The authors worked on parts of this project while they were at the Erwin Schrödinger Institute, Vienna, Austria, during the program in Modern Methods of Time-Frequency Analysis. They want to thank ESI for its support and the organizers of the program for providing an inspiring research atmosphere.

10


References [1] R. Beals, Lp and H¨ older estimates for pseudodifferential operators: necessary conditions, ”Harmonic analysis in Euclidean spaces,” Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978, Part 2, pp. 153–157, ´ Bényi, L. Grafakos, K. Gr¨ [2] A. ochenig, and K. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harm. Anal. 19.1 (2005), 131–139. ´ Bényi, K. Gr¨ [3] A. ochenig, C. Heil, and K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Oper. Theory 54.2 (2005), 387–399. ´ Bényi, K. Gr¨ [4] A. ochenig, K. A. Okoudjou, and Luke Rogers, Unimodular Fourier multipliers for modulation spaces, preprint. ´ Bényi and R. H. Torres, Almost orthogonality and a class of bounded bilinear pseudodiffer[5] A. ential operators, Math. Res. Letters 11.1 (2004), 1–12. [6] A. P. Calder´ on and R. Vaillancourt, On the boundedness of pseudodifferential operators, J. Math. Soc. Japan 23 (1971), 374–378. [7] H. G. Feichtinger, Modulation spaces on locally Abelian groups, in: “Wavelets and Applications,” (Chennai, 2002), M. Krishna, R. Radha and S. Thangavelu, eds., Allied Publishers, New Delhi (2003), 99–140. [8] H. G. Feichtinger and G. Narimani, Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., to appear. [9] K. Gr¨ ochenig, and C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, 34 (1999), no. 4, 439–457. [10] E. Galperin and K. Gr¨ ochenig, Uncertainty principles as embeddings of modulation spaces, J. Math. Anal. Appl. 271 (2002), 181–202. [11] K. Gr¨ ochenig, “Foundations of Time-Frequency Analysis,” Birkhäuser, Boston, 2001. [12] K. Gr¨ ochenig, An uncertainty principle related to the Poisson summation formula, Studia Math. 121 (1996), 87–104. [13] C. Heil, J. Ramanathan, and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal. 150 (1997), no. 2, 426–452. [14] J. A. Hogan and J. D. Lakey, Embeddings and uncertainty principles for generalized modulation spaces, in: “Modern Sampling Theory,” J. J. Benedetto and P. J. S. G. Ferreira, eds., Birkh¨ auser, Boston (2001), 73–105. [15] L. H¨ ormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140. [16] W. Littman, The wave operator and Lp norms, J. Math. Mech. 12 (1963), 55-68. [17] K. A. Okoudjou, Embeddings of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc. 132 (2004), 1639–1647. [18] J. Sj¨ ostrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1(2) (1994), 185–192. [19] T. Tao, Local and global analysis of nonlinear dispersive and wave equations, NSF-CBMS Lecture Notes, New Mexico State University, June 2005. [20] T. Tao, http://www.math.ucla.edu/∼tao/dispersive. [21] J. Toft, Modulation spaces and pseudodifferential operators, J. Funct. Anal. 207 (2004), 399– 429. [22] J. Toft, Covolutions and embeddings for weighted modulation spaces, in: ”Advances in pseudodifferential operators,” Oper. Theory Adv. Appl., 155, Birkhäuser, Basel 2004, 165–186.


11

´ a ´d Be ńyi, Department of Mathematics, 516 High St, Western Washington UniArp versity, Bellingham, WA 98225, USA E-mail address: [email protected] Kasso A. Okoudjou, Department of Mathematics, University of Maryland, College Park, MD 20742, USA E-mail address: [email protected]